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Properties

Label 1-189-189.131-r0-0-0
Degree $1$
Conductor $189$
Sign $-0.970 + 0.240i$
Analytic cond. $0.877712$
Root an. cond. $0.877712$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.173 + 0.984i)5-s + (0.5 − 0.866i)8-s − 10-s + (−0.173 + 0.984i)11-s + (−0.766 + 0.642i)13-s + (0.766 + 0.642i)16-s + 17-s − 19-s + (0.173 − 0.984i)20-s + (−0.939 − 0.342i)22-s + (−0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.173 + 0.984i)5-s + (0.5 − 0.866i)8-s − 10-s + (−0.173 + 0.984i)11-s + (−0.766 + 0.642i)13-s + (0.766 + 0.642i)16-s + 17-s − 19-s + (0.173 − 0.984i)20-s + (−0.939 − 0.342i)22-s + (−0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (−0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.970 + 0.240i$
Analytic conductor: \(0.877712\)
Root analytic conductor: \(0.877712\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ -0.970 + 0.240i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09404033929 + 0.7698440336i\)
\(L(\frac12)\) \(\approx\) \(0.09404033929 + 0.7698440336i\)
\(L(1)\) \(\approx\) \(0.5737101165 + 0.5820536268i\)
\(L(1)\) \(\approx\) \(0.5737101165 + 0.5820536268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (-0.173 + 0.984i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + T \)
97 \( 1 + (0.939 - 0.342i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.97900242915593532379008697217, −25.91740730046369784133732025951, −24.72573592675360644335428595302, −23.79384571292327539172559660622, −22.70162942894005852065836746451, −21.57546182326861210043511006802, −20.989840333548277955057811064530, −19.97331183375947610300369905735, −19.20577850789276333445391349983, −18.122949799667458185366635355063, −17.03044574232660986075763531270, −16.3694505226357593412838921380, −14.70892203903030900092650812322, −13.57953864044603874044553042362, −12.687224028460199946962423372833, −11.95101903711408606260836608285, −10.66231841432307202356362290932, −9.758245327038054090447713154888, −8.6323029296397140823339356591, −7.88710647206958648794637233966, −5.79904742268351574151206450941, −4.79598861215057026409969652699, −3.53083684208853759587237950669, −2.15565764564460608624410655734, −0.64780529878134612998349549229, 2.062369950073575140179787550713, 3.80849455564796134332284594409, 5.07947961469182332160486774103, 6.34824096539880235699207924891, 7.18055887849534474413491399852, 8.113820623739653660932413370207, 9.69533373787082249950603040495, 10.15841203893096752194832473573, 11.75640701327158665900820867703, 13.05528581514332702837974301756, 14.26775103211537512226128464343, 14.81947848395320164442375312021, 15.77350308070649094006860246076, 17.05666126829561902577927348213, 17.707414948930650468265126715263, 18.79429385591913748238470788133, 19.44667876834707186831667648197, 21.118822233029588222818322742429, 22.12553702405982771496211026429, 22.97758772730478425840508373490, 23.71404378404196811460711178286, 24.88900231690664063351312567095, 25.83602842350076067642107996880, 26.27224222972690417283647944745, 27.40969122603991742271924151852

Graph of the $Z$-function along the critical line